Warning 4.7.6.10. The collection of minimal $\infty $-categories has poor closure properties:
If $\operatorname{\mathcal{C}}$ is a minimal $\infty $-category and $K$ is a simplicial set, then the $\infty $-category $\operatorname{Fun}(K, \operatorname{\mathcal{C}})$ need not be minimal (even in the case $K = \Delta ^1$).
If $\operatorname{\mathcal{C}}$ is a minimal $\infty $-category and $q: K \rightarrow \operatorname{\mathcal{C}}$ is a diagram, then the $\infty $-categories $\operatorname{\mathcal{C}}_{/q}$ and $\operatorname{\mathcal{C}}_{q/}$ need not be minimal (even in the case $K = \Delta ^0$).
If $\operatorname{\mathcal{C}}$ is a minimal $\infty $-category and $\operatorname{\mathcal{D}}$ is equivalent to $\operatorname{\mathcal{C}}$, then $\operatorname{\mathcal{D}}$ need not be minimal.